Cartesian oval

In geometry, a Cartesian oval is a plane curve consisting of points that have the same linear combination of distances from two fixed points (foci). These curves are named after French mathematician René Descartes, who used them in optics. Let P and Q be fixed points in the plane, and let d(P, S) and d(Q, S) denote the Euclidean distances from these points to a third variable point S. Let m and a be arbitrary real numbers. Then the Cartesian oval is the locus of points S satisfying d(P, S) + m d(Q, S) = a. The two ovals formed by the four equations d(P, S) + m d(Q, S) = ± a and d(P, S) − m d(Q, S) = ± a are closely related; together they form a quartic plane curve called the ovals of Descartes. In the equation d(P, S) + m d(Q, S) = a, when m = 1 and a > d(P, Q) the resulting shape is an ellipse. In the limiting case in which P and Q coincide, the ellipse becomes a circle. When it is a limaçon of Pascal. If and the equation gives a branch of a hyperbola and thus is not a closed oval. The set of points (x, y) satisfying the quartic polynomial equation where c is the distance between the two fixed foci P = (0, 0) and Q = (c, 0), forms two ovals, the sets of points satisfying the two of the following four equations that have real solutions. The two ovals are generally disjoint, except in the case that P or Q belongs to them. At least one of the two perpendiculars to PQ through points P and Q cuts this quartic curve in four real points; it follows from this that they are necessarily nested, with at least one of the two points P and Q contained in the interiors of both of them. For a different parametrization and resulting quartic, see Lawrence. As Descartes discovered, Cartesian ovals may be used in lens design. By choosing the ratio of distances from P and Q to match the ratio of sines in Snell's law, and using the surface of revolution of one of these ovals, it is possible to design a so-called aplanatic lens, that has no spherical aberration.